System Modeling and Simulation Carey Williamson Department of - PowerPoint PPT Presentation

CPSC 531: System Modeling and Simulation Carey Williamson Department of Computer Science University of Calgary Fall 2017

Outline  Probability and random variables — Random experiment and random variable — Probability mass/density functions — Expectation, variance, covariance, correlation  Probability distributions — Discrete probability distributions — Continuous probability distributions — Empirical probability distributions 2

Random Experiment  3

Probability of Events  4

Joint Probability  5

Independent Events  6

Mutually Exclusive Events  7

Union Probability  8

Conditional Probability  9

Types of Random Variables  Discrete — Random variables whose set of possible values can be written as a finite or infinite sequence — Example: number of requests sent to a web server  Continuous — Random variables that take a continuum of possible values — Example: time between requests sent to a web server 10

Probability Density Function (PDF)  d  f ( x ) F ( x ) dx 11

Cumulative Distribution Function (CDF)  12

Expectation of a Random Variable   n  x p ( x ) discrete X   i i     i 1  E [ X ]    xf ( x ) dx continuous X     13

Properties of Expectation  14

Misuses of Expectations  Multiplying means to get the mean of a product  E [ XY ] E [ X ] E [ Y ]  Example: tossing three coins — X : number of heads — Y : number of tails — E[X] = E[Y] = 3/2  E[X]E[Y] = 9/4 — E[XY] = 3/2  E[XY] ≠ E[X]E[Y]  Dividing means to get the mean of a ratio   X E [ X ]  E     Y E [ Y ] 15

Variance of a Random Variable  16

Variance of a Random Variable  Variance : The expected value of the square of distance between a random variable and its mean   2 V [ X ]  n    2  ( x ) p ( x ) discrete X  i i      i 1 2  E [( X ) ]      2 ( x ) f ( x ) dx continuous X   where, μ = E[X]    Equivalently: σ 2 = E[X 2 ] – (E[X]) 2 17

Properties of Variance  18

Coefficient of Variation  3 / 4 1   CV 3 / 2 3 19

Covariance  20

Covariance  x y xy p(x) 0 3 0 1/8 1 2 2 3/8 2 1 2 3/8 3 0 0 1/8 xy p(xy) 0 2/8 2 6/8 21

Correlation  Negative linear Positive linear correlation correlation No correlation -1 0 +1 22

Autocorrelation  Negative linear Positive linear correlation correlation No correlation -1 0 +1 23

Demo: Correlation and Autocorrelation  Correlation (if desired) can be induced by sharing or re-using random numbers between two (or more) random variables  Example: height and weight of medical patients  Example: a coin that remembers some of its recent history Negative linear Positive linear correlation correlation No correlation -1 0 +1 24

Geometric Distribution  25

Example: Geometric Distribution Geometric distribution PMF Geometric distribution CDF

Uniform Distribution  PDF CDF 27

Uniform Distribution Properties  28

Exponential Distribution  29

Example: Exponential Distribution Exponential distribution PDF Exponential distribution CDF

Light Bulb Testing (1 of 5)  Scenario: Walmart has a giant bin of lightbulbs on sale. You buy one and bring it home for testing and observation.  Assume: All light bulbs last exactly 100 hours.  Observation: Your light bulb has worked for 70 hours.  Question: How much longer is it expected to last?  Answer: 30 hours 31

Light Bulb Testing (2 of 5)  Scenario: Walmart has a giant bin of lightbulbs on sale. You buy one and bring it home for testing and observation.  Assume: Half of the light bulbs last exactly 50 hours, while the other half last exactly 150 hours. The mean is 100 hours.  Observation: Your light bulb has worked for 70 hours.  Question: How much longer is it expected to last?  Answer: 80 hours 32

Light Bulb Testing (3 of 5)  Scenario: Walmart has a giant bin of lightbulbs on sale. You buy one and bring it home for testing and observation.  Assume: Half of the light bulbs last exactly 50 hours, while the other half last exactly 150 hours. The mean is 100 hours.  Observation: Your light bulb has worked for 40 hours.  Question: How much longer is it expected to last?  Answer: 60 hours 33

Light Bulb Testing (4 of 5)  Scenario: Walmart has a giant bin of lightbulbs on sale. You buy one and bring it home for testing and observation.  Assume: Light bulbs have a working duration that is uniformly distributed (continuous) between 50 hours and 150 hours. The mean is 100 hours.  Observation: Your light bulb has worked for 70 hours.  Question: How much longer is it expected to last?  Answer: 40 hours 34

Light Bulb Testing (5 of 5)  Scenario: Walmart has a giant bin of lightbulbs on sale. You buy one and bring it home for testing and observation.  Assume: Light bulbs have a working duration that is exponentially distributed with a mean of 100 hours.  Observation: Your light bulb has worked for 70 hours.  Question: How much longer is it expected to last?  Answer: 100 hours 35

Memoryless Property  36

Example: Exponential Distribution  37